This might seem kind of a dumb question...But, I was wondering if anyone could guide on when to use the summation of Poisson random variables.

For example, in SOA Sample #216:
"Company XYZ provides a warranty on a product that it produces. Each year, the number of warranty claims follows a Poisson distribution with mean c. The probability that no warranty claims are received in any given year is 0.60.
Company XYZ purchases an insurance policy that will reduce its overall warranty claim payment costs. The insurance policy will pay nothing for the first warranty claim received and 5000 for each claim thereafter until the end of the year.
Calculate the expected amount of annual insurance policy payments to Company XYZ."

Okay, so, I understand why summation was used for the number of warranty claims.

However, in SOA Sample #213: "The number of traffic accidents per week at intersection Q has a Poisson distribution with mean 3. The number of traffic accidents per week at intersection R has a Poisson distribution with mean 1.5.
Let A be the probability that the number of accidents at intersection Q exceeds its mean.
Let B be the corresponding probability for intersection R.
Calculate B Ė A."

This might seem kind of a dumb question...But, I was wondering if anyone could guide on when to use the summation of Poisson random variables.

For example, in SOA Sample #216:
"Company XYZ provides a warranty on a product that it produces. Each year, the number of warranty claims follows a Poisson distribution with mean c. The probability that no warranty claims are received in any given year is 0.60.
Company XYZ purchases an insurance policy that will reduce its overall warranty claim payment costs. The insurance policy will pay nothing for the first warranty claim received and 5000 for each claim thereafter until the end of the year.
Calculate the expected amount of annual insurance policy payments to Company XYZ."

Okay, so, I understand why summation was used for the number of warranty claims.

However, in SOA Sample #213: "The number of traffic accidents per week at intersection Q has a Poisson distribution with mean 3. The number of traffic accidents per week at intersection R has a Poisson distribution with mean 1.5.
Let A be the probability that the number of accidents at intersection Q exceeds its mean.
Let B be the corresponding probability for intersection R.
Calculate B Ė A."

I think we need you to elaborate on your confusion more to better help you. Both questions are asking for something that is calculated with sums.

The first question asks for the expected value of a transformation of a familiar random variable which can be broken into 5000*(the mean)-5000*(a certain probability). Both the mean and this probability are calculated as sums. The probability can either be calculated as an infinite sum or as 1 minus a finite sum (its complement). The expected value can be written as a sum and calculated with infinite sum convergence rules or can just be written because you already solved for the parameter of this Poisson which is that number.

The second problem asks you for the difference between A and B, which are both defined as probabilities for two different Poissons. A is defined as P(Q>3) since it's given as "the probability that Q exceeds its mean" and its mean is known to be 3. Similarly, B is defined as P(R>1.5). Both of these can be calculated as infinite sums:
A=P(Q=4)+P(Q=5)+...
B=P(R=2)+P(R=3)+...
Or can be written as 1 minus their complements:
A=1-[P(Q=0)+P(Q=1)+P(Q=2)+P(Q=3)]
B=1-[P(R=0)+P(R=1)]

I think we need you to elaborate on your confusion more to better help you. Both questions are asking for something that is calculated with sums.

The first question asks for the expected value of a transformation of a familiar random variable which can be broken into 5000*(the mean)-5000*(a certain probability). Both the mean and this probability are calculated as sums. The probability can either be calculated as an infinite sum or as 1 minus a finite sum (its complement). The expected value can be written as a sum and calculated with infinite sum convergence rules or can just be written because you already solved for the parameter of this Poisson which is that number.

The second problem asks you for the difference between A and B, which are both defined as probabilities for two different Poissons. A is defined as P(Q>3) since it's given as "the probability that Q exceeds its mean" and its mean is known to be 3. Similarly, B is defined as P(R>1.5). Both of these can be calculated as infinite sums:
A=P(Q=4)+P(Q=5)+...
B=P(R=2)+P(R=3)+...
Or can be written as 1 minus their complements:
A=1-[P(Q=0)+P(Q=1)+P(Q=2)+P(Q=3)]
B=1-[P(R=0)+P(R=1)]

So I see sums in both problems. Does this clear things up for you?

Yes!!! It does make a lot more sense! Thank you so much! Sorry, should have have written my answer more clearly! But, your reply was very helpful! Thank you so much!